PACS: 02.20.-a, 61.50.Ah 1. Introduction the vectors an = ma3 (n1, n2, m are integers). The set of elements The subperiodic rod groups R are the 75 three-dimen(E|a(2))(gi|vi + am)(1) n sional groups with one-dimensional translations which turn contains a group of three-dimensional translations up to be in concomitant relationships with three-dimensional (E|a(2) + am) T and is some space group provided the n space groups G . Rod groups describe the symmetry translational symmetry (the group T ) is compatible with of one-periodic systems and can be used for studying the point symmetry F of the rod group R. This condition is polymeric molecules, nanotubes and others similar objects.
fulfilled if the vector a3 is perpendicular to the plane of Besides, this geometrical symmetry appears when applying the translations a(2). Indeed the translations ma3 are coma uniform magnetic field on bulk crystals, superlattices, n patible with F as they are elements of R. The compatibility quantum wells . Irreducible representations (IRs) of rod (2) groups are necessary for physical applications (e. g., deriving of the translations a(2) T with point group F follows n selection rules for optical transitions). from the fact that the rotations (proper and improper) A subperiodic rod group R can contain the following from R transform the rod into itself and, therefore, any elements: translations in one direction (of a vector d); vector perpendicular to the rod — into the vector also two-, three-, four- or six-fold rotation or screw axes perpendicular to the rod. Thus the set of elements (1) pointed in this direction; two-fold axes perpendicular to forms one of three-periodic space groups G which has it; reflection planes containing d; reflection planes per- the same point symmetry as the rod group R. Moreover, (2) pendicular to d. Every subperiodic rod group R is in the translational group T is invariant in G: along with one-to-one correspondence with some three-periodic space the translation (E|a(2)) it contains also the translation n group G: it is a subgroup of G (R G) and has the (E|gia(2)) =(gi|vi + am)(E|a(2))(gi|vi + am)-1 for any gi n n same point symmetry group. To obtain a rod group R, from (1). The group G may be represented as a semi-direct it is sufficient to keep translations only in one direction (2) product of T and R in a related space group G. These groups (R and G) (2) (2) have the same international notations. For example, G G = T R, G = (gi|vi + am)T. (2) 143 C1 (P3) R 42 (p3); G 173 C6 (P63) R 56 (p63).
i 3 The IRs of rod groups R may be generated in the same For some rod groups (R 1, R 2, R 4, R 5) of low point way as for three-periodic space groups G. All the IRs of R symmetry, the plane may be inclined with respect to the are contained in the IRs of the related space group G and (2) vector a3. In this case, the translational group T remains can be taken directly from, e. g., Tables of Ref. . The invariant in G. A rod group R is a subgroup of G and procedure how to make this is given in Section 2.
(2) isomorphous to the factor group G/T. According to the little group method ([4,5], see also Appendix) every IR of R 2. The relation between IRs of space is related to a definite IR of G of the same dimension. In (2) and subperiodic rod groups these IRs of G all the elements of the coset (gi|vi + am)T are mapped by the same matrix. In particular, all the (2) (2) Let (gi|vi + am) R be elements of a rod group R, translations in T (coset (E|0)T ) are mapped by unit where gi is a proper or improper rotation followed by matrices.
improper translation vi and am = ma3 are lattice translations Let us choose, in the space of an IR of G, the basis which (2) of R. Consider a group T of two-dimensional translations is at the same time the basis of the IRs of its invariant (2) (2) a(2) = n1a1 + n2a2 in the plane which does not contain subgroup T. Then the translations belonging to T n 1296 V.P. Smirnov, P. Tronc are mapped by the diagonal matrices with the elements Table 1. Single- ( ) and double-valued ( ) IRs 7- 1- of the rod group R 56 (p63) at the point (k = 0) of the oneexp(-ik(3) · a(2)). These matrices become the unit ones, n dimensional BZ ( =(0, 0, c/2), exp(i/6)) if at any integers n1 and n exp(-ik(3) · a(2)) =1. (3) Element = = = = = n 1 3 5 4 6 7 12 8 11 9 1 1 1 1 -1 -1 -This condition holds for any k(3) = K3 in the direction of the basic translation vector K3 = a1 a2 (C6|) 1 -1 -i i - -i Va (C3|0) 1 1 i i i i -of the three-dimensional Brillouin zone (BZ) of the (C2|) 1 -1 1 -1 i -i i space group G, which is perpendicular to the plane.
(C2|0) 1 1 -i -i i i The only primitive translation vector K = a and all (C5|) 1 -1 i -i - -i |a|the wave vectors k = K(-1/2 < 1/2) in the onedimensional BZ of the rod group R are directed along the Table 2. Single- (A1-A6) and double-valued (A7-A12) IRs vector a = a3. The correspondence between k(3) and k of the rod group R 56 (p63) at the point A (k = /c) of the oneis established by the transformation law of basic vectors dimensional BZ ( =(0, 0, c/2), exp(i/6)) of IRs under translation operations an of the rod group:
exp(-ik(3) · a3) =exp(-ik · a), i. e. =. If a then Element A1 = A2 A3 = A6 A4 = A5 A7 = A11 A8 = A12 A9 Ak = k(3), otherwise k is the projection of k(3) on the 1 1 1 -1 -1 -1 -direction of a = a3.
(C6|) -i - i -i 1 -The star of any vector k(3) lies entirely in the direction (C3|0) 1 i i i i -1 -of the primitive vector K3. Therefore the correspondence (C2|) -i -i i -1 1 -1 of IRs mentioned above takes place both for allowed IRs of (C2|0) 1 -i -i i i 1 little groups Gk(3) (in G) and Rk (in R) and for the full IRs (C5|) -i - -i i 1 -of G and R. So the subduction of any small IR of a little group Gk(3) (full IR of G with wave vector star k(3)) on the elements of the rod group R generates some small IR of the directly from Tables of Ref.  for G = C1 space group.
little group Rk (full IR of R with the wave vector star k) The group C1 is symmorphic. The IRs with k on the line of the same dimension.
In Tables of IRs of space groups, one finds usually small A for the elements (C3|ma) differ from those for (C3|0) by (3) the factor exp(-ik · ma) as this factor corresponds to the IRs of little groups Gk ( see, e. g., Ref. ). An IR d(k,)(g) translation ma. Another example is the non-symmorphic of a little group Gk G is at the same time an IR d(k,)(g) rod group R 56 (p63). Its IRs are related to the IRs of of a little rod group Rk R with k = k(3), when a, the non-symmorphic space group G 173 (C6). This is the or k being projection of k(3) on the direction of a = a3. geometrical symmetry of bulk materials with the wurtzite The analogous procedure of IRs generation is valid for structure (e. g. bulk GaN) and the superlattices of the IRs of 80 three-dimensional groups with two-dimensional (GaN)m(AlN)n type with odd values of m + n (the Ctranslations (layer) groups . 6v non-symmorphic space group), when the magnetic field B is directed along the symmetry axis. Since the crystal 3. Discussion system is the same as in the first example (hexagonal lattice), one has also k(3) = k and takes the IRs of R To illustrate the proposed procedure let us consider for point (the center of one-dimensional BZ, Table 1) semiconductor structures under a magnetic field. Let us and A (the edge of one-dimensional BZ, Table 2) directly consider the symmetry of bulk semiconductors with the zinc from Tables of Ref.  for G = C6 space group. Note blende structure (the Td(2) symmorphic space group), such that all the points in the BZ of the rod group R 56 have as the GaAs or AlAs crystals for example, under a magnetic the same point symmetry C6. The IRs with k on the field B parallel to the symmetry axis C3, or superlattices line A for the elements (C6|a/2 + ma) differ by the factor of the (GaN)m(AlN)n type with an even value of m + n exp -ik · (m + 1/2)a from those for element (C6|a/2) (the C1 symmorphic space group), when the magnetic 3v at (k = 0) as it follows from the theory of projective field B is directed along the symmetry axis C3. These representations.
systems have the geometrical symmetry described by the rod group R 42 (p3), whose IRs are related to those of the space group G 143 (C1). In this case the plane of 4. Selection rules for electrical dipole the lattice translations a(2) = n1a1 + n2a2 is perpendicular transitions n to the translation vector a of the rod group which coincides with lattice translation vector a3 of G. Thus k(3) = k. The stationary states of a system with the symmetry of a One takes the IRs of R for point (the center of one- rod group R are classified according to the small IRs |k, dimensional BZ) and A (the edge of one-dimensional BZ) of the little group Rk R.
Физика твердого тела, 2006, том 48, вып. Irreducible representations of subperiodic rod groups Table 3. Direct (Kronecker) products (Ai Aj and A Aj) of the single- (A1-A6) and double-valued (A7-A12) IRs at A-point of the BZ j for rod group R 56 (p63) IR A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A A2 A2 1 4 3 6 5 7 8 9 10 11 A1 A1 2 3 4 5 6 8 7 10 9 12 A6 A4 3 6 5 2 1 9 10 11 12 7 A5 A3 4 5 6 1 2 10 9 12 11 8 A4 A6 5 2 1 4 3 11 12 7 8 9 A3 A5 6 1 2 3 4 12 11 8 7 10 A11 A7 8 9 10 11 12 3 4 5 6 1 A12 A8 7 10 9 12 11 4 3 6 5 2 A9 A9 10 11 12 7 8 5 6 1 2 3 A10 A10 9 12 11 8 7 6 5 2 1 4 A7 A11 12 7 8 9 10 1 2 3 4 5 A8 A12 11 8 7 10 9 2 1 4 3 6 Not e. =, =, =, =, =.
3 5 4 6 7 12 8 11 9 Let us consider the selection rules  for transitions selection rules, where the operator P has the symmetry between stationary states of symmetry |k( f ), ( f ) and of phonon participating in the transition. In GaN crystal, |k(i), (i) caused by an operator P(k(p), (p)) transforming atoms occupy the sites of b-type of symmetry C3v. Under according to the IR (k(p), (p)) of R. If the operator P the magnetic field B directed along the symmetry axis, the transforms according some reducible rep of R, one can symmetry of the system reduces down to rod group R 56, consider the selection rules for every of its irreducible and the site symmetry of atoms down to C3. In this case components separately. the symmetries of phonons are given by representations The transition probability is governed by the value of the of rod group R 56 induced by the vector representation matrix element a + e(1) + e(2) of the site symmetry group C3. The short symbol  of this representation is (1, 4, 2, 5, 3, 6), i. e., k( f ), ( f )|P(k(p), (p))|k(i), i. (4) phonons can be of any symmetry. The short symbol determines the symmetry of phonons in all the points in The transition is referred to as allowed by symmetry, if the a one-dimensional BZ. For example, as it was established triple direct (Kronecker) product above, the electric dipole transtitions are allowed from initial electronic A8 state to the intermediate A8, A9, A(k( f ), ( f )) (k(p), (p)) (k(i), (i)) (5) states. From these states, with assistance of the phonons of contains the identity IR of R, or symmetry A3, the transitions are allowed into the final,, states (see Table 3). If the intermediate state is of 8 (k( f ), ( f )) (k(i), (i)) (k(p), (p)) = 0, (6) symmetry, the same phonon allows the transition in the finale state A12.
i. e., it is necessary to find the direct product of two IRs of the rod group R (complex conjugate IRs are also IRs of R).
Let us take the case of GaN bulk crystal with the wurtzite 5. Conclusion structure under the magnetic field B directed along the symmetry axis (rod group R 56 (p63)). The symmetry of It is not necessary to generate IRs of rod groups R.
the electrical dipole operator in this group described by vecAs it is demonstrated above, they can be taken directly tor representation = (z ) + (x - iy) + (x + iy).
1 4 from the existing Tables of IRs for space groups with threeAs k(p) 0, k( f ) k(i), only the so-called direct transitions:
, A A, etc. are allowed (wave vector selection rules). In particular, when the spin-orbit interaction is Appendix taken into account, the symmetry of allowed final stated for A A transitions is pointed out in Table 3 by the entries Let H be an invariant subgroup of a group G (H G, of the rows containing =, =, or = in the 1 1 4 6 6 gHg-1 = H, g G) and d()(h) be an IR of H. The columns corresponding to the symmetry of the initial state.
group G can be developed in terms of left cosets with For example, the direct transitions are allowed from the respect to H initial state of symmetry A8 to final states of symmetry A8, A9 and A11.
t In the case of phonon assisted electric dipole transitions, G = g H, g1 = E (identity element). (A1) j these selection rules have to be supplemented with the j=10 Физика твердого тела, 2006, том 48, вып. 1298 V.P. Smirnov, P. Tronc The cosets g H compose a factor group G/H with compoj sition law giHg H = gig g-1Hg H = gig HH = gig H. (A2) j j j j j j The matrices d(µ)(g hg-1) form an IR of H conjugate j j to d(µ)(h) by means of g. The set of elements of j those left cosets g H (p = 1, 2,..., s t) for which p the IRs d(µ)(g hg-1) are equivalent to the IR d(µ)(h) p p (d(µ)(g hg-1) =Ad(µ)(h)A-1, where A is some non-singup p lar matrix of the same order as d(µ)(h)), forms a group Gµ G called the little group for the IR d(µ)(h) of H G [4,5]. If the IR of Gµ, when restricted to H, contains only the IR d(µ)(h) of H, it is called allowed (small). Small IRs of the little group Gµ compose a part of all the IRs of Gµ.
According to the little group method [4,5], the little group G1 for the identical IR d(1)(h) =1 (h H, all the elements are mapped by 1) of an invariant subgroup H coincides with the whole group G (G1 = G). Then there is a simple relation between the allowed IRs of the group G1 = G and the IRs of the factor group G/H: every IR of G/H generates some allowed IR of G, in which all the elements of the coset giH in the decomposition (A1) are mapped by the same matrix, namely by the matrix of the factor-group G/H IR for the coset giH.
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